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Introduction to

Quantum Computing

Petros Wallden∗

Lecture 1: Introduction

18th September 2017

∗School of Informatics, University of Edinburgh

Resources

1. “Quantum Computation and Quantum In-

formation”

by Michael A. Nielsen & Isaac L. Chuang

2. Lecture Notes available on

http://qcintro.wordpress.com

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Moore’s Law

& Quantum Mechanics

• The number of transistors in each microchip doubleevery two years

• Soon we will reach atomic scale

• Quantum Mechanics govern physical systems at thisscale

• Quantum Fluctuations and Uncertainty will affectclassical computations

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Bits Vs Qubits

Bit QubitTakes values either 0 or 1 Can behave as being simul-

taneously 0 and 1: α |0〉+β |1〉

Measurement reveals thevalue of the bit

Measurement disturbs thesystem

Can be copied Cannot be copiedString of bits are describedin terms of single bits (lo-cal)

String of qubits can haveproperties that cannot bedescribed in terms of singlequbits (non-local)Qubits behave as wavesand interfere with eachother

• Qubits are physical systems. Many different systemshave been used such as:

Photons (polarization, number, time-bin encoding), Co-

herent Light, Electrons (spin, number), Nuclear spin,

Optical lattices, Superconductors, etc.

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Quantumness as ResourceNobel laureate Richard Feynman 1982:

Quantum Computer is a computer that uses QM to itsadvantage. It can simulate quantum systems.

Great Developments:

- Quantum Algorithms can lead to speed-up

- Quantum Computers can break classical Cryptosys-tems such as the RSA

- Quantum Cryptogaphy can encrypt messages with Un-conditionally Security (not relying in computational as-sumptions)

- Principles of Quantum Computation can be used to

simulate and explore physical phenomena at domains

that are not accessible from Black Hole thermodynamics

to Condense Matter Physics

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Secure Quantum Communication

- Many quantum cryptographic protocols: Encryption,secret sharing, digital signatures, coin flipping, Uncon-ditionally secure homomorphic encryption

- Implementations of Quantum Key Distribution Net-works between cities exist in many countries. QKDsystems are provided by commercial companies (e.g.idQuantique)

Quantum Computers

- There exist different models of Quantum Computa-tion: Quantum Circuit, Measurement Based (these twowill be covered), Adiabatic QC, Topological QC

- Implementations have attempted to used different phys-ical systems. Still not scalable (only few qubits op-erations e.g. factored 143). Superconductor based,Trapped ion, Optical lattices, Nuclear magnetic reso-nance, quantum optics

NQIT (Networked Quantum Information Technologies)Hub (lead by Oxford, Edinburgh is part of): Q20:20, 20ion traps of 20 qubits each, connected with photons.

Quantum Algorithms

Speed-up

- 1985 Deutsch & Jozsa showed the first speed up

Given a Boolean function f : {0,1}n → {0,1} determineif it is constant or balanced

|f〉 = 1√2n

∑x∈{0,1}n(−1)f(x) |x〉

The state for any constant function is orthogonal to thestate of any balanced function

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- 1994 Simon’s Problem

Given a function f : {0,1}n → {0,1}n finds a such thatf(x+ a) = f(x)

- 1994 Shor’s Algorithm

Given n-bit integer, find the prime factorisation. Breaksthe RSA cryptosystem (most currently used public keyencryptions are based on this)

History• 1980s Idea of quantum computation. Paul Benioff,

Yuri Manin, Richard Feynman, David Deutsch

• 1990s Theory of efficient quantum simulation. SethLloyd

• 1994 Peter Shor’s algorithms for factoring and dis-crete log. Quantum computers can break RSA,Diffie-Hellman, El Gamal, Elliptic Curve Cryptog-raphy and others

• 2001 Experiment factors 15 using Shor’s algorithm

• 2010s D-Wave, Google, IBM, NQIT and variousuniversities work on developing quantum computers

How serious is the involvement in quantum computa-tion?

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Who invests in

Quantum Computing?

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Who invests in

Quantum Computing?

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Applications

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Misconceptions

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Misconceptions

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State of Art• 2001 Shor’s algorithm factors 15 on 7 qubits

• 2011 Shor’s algorithm factors 21

• 2012 Universal quantum computation on 2 faulttolerant qubits

• 2014-2015 Qubits and gates in silicon chips

• 2015 D-Wave 2X, 1000 qubits, optimization prob-lems, no fault tolerance

• 2016 IBM, universal quantum computation on 5fault tolerant qubits (publicly available)

• 2020 NQIT, Q20:20, fault tolerant (20 qubits),scalable

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State of Art:

Cryptography

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State of Art:

Cryptography

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What can you buy

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What can you buy

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Quantum MechanicsNobel laureate Niels Bohr (photo with Einstein)

“Anyone who is not shocked by quantum theory has notunderstood it”

- Basic resource for QC is the distinct properties ofquantum theory

- To appreciate this one needs to (attempt to) under-stand QM

- QM has been proven very successful and all so fartested predictions has been verified at a unprecedentlevel of accuracy

- However, the conceptual challenges posed by QM areprofound. Classical notions such as locality, non-contextuality,determinism even realism has been challenged

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- The role of the observer and of the measurement arevery different

- Properties with no classical analogue: Uncertainty,wave-particle duality, no-cloning, indistinguishability ofquantum states, teleportation

- Also QM is incompatible with the other most success-

ful physical theory General Relativity. This is possible

because the former deals with the micro-world while the

latter with macro-world. However, for a complete theory

of nature one needs to construct a theory that includes

both QM and GR and this is probably the greatest chal-

lenge for contemporary physics.

Content of the Course• Basic concepts from Linear Algebra

• Axioms of Quantum Mechanics

• Non-locality, Bell’s inequalities and the interpreta-tions of QM

• No-cloning and no-deleting theorem

• Quantum Computing via the circuit model

• Quantum complexity

• Quantum Algorithms

• Quantum Cryptography

• Quantum Computing via the measurement-basedmodel

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